How do I statistically evaluate a ML model?
There are a lot of ways to evaluate the performance of an ML model.  You mentioned AUROC and AUPRC.  Generally you start with the confusion matrix and derive metrics such as sensitivity, accuracy, recall, precision, etc.  You can see a good outline of them here.  

It seems what you are asking is a shortcut to determining how good your sentiment classification model is but there aren't any without labeled test data.  You either do this by hand or you find a test set in the world, preferably something that is well know and documented and also fits your objectives.  I recommend you read Neil Slater's answer at https://datascience.stackexchange.com/questions/12226/how-do-i-assess-which-sentiment-classifier-is-best-for-my-project/12228.  He gives some good advice on the subjectivity of sentiment analysis classification and points out a labeled data set of Tweets which you might be able to use to test your classifier.

I also found this Kaggle competition which has a test set that might be of help to you: Angry Tweets